2015/03/10 by Jean‐François Biasse, Jean-Francois Biasse, Biasse, Jean-Francois
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.CR #math.NT
paper · pdf · doi:10.48550/arxiv.1503.03107
openalex publication_date 2015/03/10 · arxiv created 2017/03/24 · arxiv updated 2017/03/28 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
We present a heuristic algorithm to compute the ideal class group, and a generator of a principal ideal in ℚ(ζps) in time 2^O(n1/2+ε) for n:= deg(K) and arbitrarily small ε. This yields an attack on the schemes relying on the hardness of finding a short generator of a principal ideal such as such as the homomorphic encryption scheme of Vercauteren and Smart, and the multilinear maps of Garg, Gentry and Halevi. We rely on the work from Cramer, Ducas, Peikert and Regev. They proved that finding a short generator polynomially reduces to finding an arbitrary one. The complexity is better than when we rely on the work of Biasse and Fieker on the PIP, which yields an attack in time 2^n2/3+ε for arbitrarily small ε >0. Since Sep. 30 2016 We present practical improvements to our methods. Moreover, we describe a variant that solves the PIP on input ideal I of norm less than 2nb in time 2^O(nc+o(1)) for 2/5 < c < 1/2 and b≤ 7c -2 given a one time precomputation of cost 2^O(n2-3c+ε) for an arbitrarily small ε. This also solves γ-SVP in principal ideals of ℚ(ζps) for γ∈ e^O(√(n)). On principal ideals of norm less than 2nb, we can leverage the precomputation to achieve a better asymptotic run time than the BKZ algorithm.